Abstract not found

Abstract. The homotopy theory of the suspensions of the real projective plane is largely investigated. The homotopy groups are computed up to certain range. The decompositions of the self smashes and the loop spaces are studied with some applications to the Stiefel manifolds.

In (5), we and Fred Cohen gave some quite general splitting theorems. These described how to decompose the suspension spectra of certain filtered spaces CX as wedges of the suspension spectra of their successive filtration quotients Dq X. The spaces CX were of the form Cr xIr/ ( ~) for suitable sequences of spaces {Cf} and {Xr}, and the construction CX was intended to be a reworking in 'proper generality ' of the constructions introduced in (9). We no longer believe that sufficient generality was achieved in (5). The original motivation concerned iterated loop spaces Q. n T, n X, where X is path connected. Consider a cofibration i: A->X, where A is also connected. Let 8: X/A^-C(i)->'LA be the standard map and consider the fibre Fn(X, A) of Q n- 1 S m ~ 1 (3) (n> 1). By a slight elaboration of the argument in (9), §6, which gives complete details when X is the cone on A, there is a commutative diagram En (X. A) Fn (X, A) where the bottom row is the canonical fibration. By (9), 7-3, the top row is a quasifibration.